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relation_algebras [2010/09/17 18:16]
jipsen
relation_algebras [2010/09/17 20:45] (current)
jipsen
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Abbreviation: **RA** Abbreviation: **RA**
====Definition==== ====Definition====
-A \emph{relation algebra} is a structure $\mathbf{A}=\langle A,\vee,0,\wedge,1,\neg,\circ,^{\smallsmile},e\rangle$ such that+A \emph{relation algebra} is a structure $\mathbf{A}=\langle A,\vee,0,\wedge,1,\neg,\circ,^{\smile},e\rangle$ such that
$\langle A,\vee,0,\wedge,1,\neg\rangle$ is a [[Boolean algebra]] $\langle A,\vee,0,\wedge,1,\neg\rangle$ is a [[Boolean algebra]]
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====Examples==== ====Examples====
-Example 1: +Example 1: $\langle \mathcal P(U^2), \cup, \emptyset, \cap, U^2, -, \circ, ^\smile, id_U \rangle$ the full relation algebra of binary relations on a set $U$. 
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 +Example 2: $\langle \mathcal P(G), \cup, \emptyset, \cap, G, -, \circ, ^\smile, \{e\} \rangle$ the group relation algebra of a [[group]] $\langle G, *, ^{-1}, e \rangle$, where $X\circ Y=\{x*y : x\in X, y\in Y\}$ and $X^\smile=\{x^{-1} : x\in X\}$.
====Basic results==== ====Basic results====
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^[[Strong amalgamation property]]  |no | ^[[Strong amalgamation property]]  |no |
^[[Epimorphisms are surjective]]  |no | ^[[Epimorphisms are surjective]]  |no |
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====Finite members==== ====Finite members====
$\begin{array}{lr} $\begin{array}{lr}
-[http://localhost/gap/ramaddux.html Small relation algebras] 
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f(1)= &1\\ f(1)= &1\\
f(2)= &1\\ f(2)= &1\\
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f(6)= &0\\ f(6)= &0\\
\end{array}$ \end{array}$
 +
 +[[http://www.chapman.edu/~jipsen/gap/ramaddux.html|Small relation algebras]]
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====Subclasses==== ====Subclasses====
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[[Square-increasing relation algebras]] [[Square-increasing relation algebras]]
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====Superclasses==== ====Superclasses====
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====References==== ====References====
-[(Ln19xx> +/*[(Ln19xx> )]*/
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